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Trigonometry In Music Theory

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Marina Sa

on 15 February 2015

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Transcript of Trigonometry In Music Theory

Trigonometry In Music Theory
Trigonometry In Music Theory
“Mathematics and music, the most sharply contrasted fields of scientific activity . . . ties together all the activities of our mind.”
-Hermann von Helmholtz
What is music?
mu'sic
'myo͞ozik/
noun

1. the art of arranging tones in an orderly sequence so as to produce a unified and continuous composition
2. In regards to mathematics it is the ratios of sound waves

As we know, there are sound waves coming from all around us, even though the average human can only hear from 20 Hz to 20,000 Hz.

Therefore, 20 Hz would have a much lower frequency or sound than 20,000 Hz.
The frequencies of the average piano range is from 27 Hz to 4186 Hz with middle C being 261.63 Hz.
Middle C, with a frequency of 262 Hz (or 262 repeating tones per second) creates a pattern that looks like this . .
Mathematically, this note, C is
referred to as:
sin (542 π x)
The note A above middle C produces a wave like this, while C# and E produce waves like these
As you can see, the higher the note, the higher the frequencies and the space between wavelengths decrease.

Every note is determined by the size of its sine wave, therefore since E has a smaller wave than A, it has a higher frequency.
But how do frequencies relate to functions such as sine, cosine, and tangent?
The unit circle also happens to correspond to wave amplitudes.
And just like how the unit circle corresponds to various frequencies, sound can also respond to various shapes, which is call cymatics.

The higher the sine function, the higher the frequency and therefore, the more intricate the pattern.
"The wave is not the water. The water merely told us about the wave moving by."
- R. Buckminster Fuller

To get a chord, simply add up all of the sine functions of each note!
Even something as seemingly free-flowing and spontaneous as music has mathematical significance and this is important to realize because it means that there are patterns which we cannot always discern, yet they are present.

So why is this important?
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